Theorem spw 2036

Description: Weak version of the specialization scheme sp 2191. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 2191 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 2191 having mutually distinct setvar variables and no wff metavariables (see ax12wdemo 2141 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 2191 are spfw 2035 (minimal distinct variable requirements), spnfw 1981 (when 𝑥 is not free in ¬ 𝜑), spvw 1983 (when 𝑥 does not appear in 𝜑), sptruw 1808 (when 𝜑 is true), spfalw 1982 (when 𝜑 is false), and spvv 1990 (where 𝜑 is changed into 𝜓). (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 27-Feb-2018.)

Hypothesis

Ref	Expression
spw.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
spw	⊢ (∀𝑥𝜑 → 𝜑)

Distinct variable groups:   𝑥,𝑦   𝜓,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem spw

Step	Hyp	Ref	Expression
1		ax-5 1912	. 2 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓)
2		ax-5 1912	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)
3		ax-5 1912	. 2 ⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑)
4		spw.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	1, 2, 3, 4	spfw 2035	1 ⊢ (∀𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782

This theorem is referenced by:  hba1w  2051  19.8aw  2054  exexw  2055  spaev  2056  ax12w  2139  rspw  3215  reldisj  4407  ralidmw  4471  dtruALT2  5317  bj-ssblem1  36899  bj-ax12w  36922  eu6w  43034